A 1-D radial vaneless diffuser model accounting for the effects ofaerodynamic blockage

Stuart, C., Spence, S., Kim, S., Filsinger, D., & Starke, A. (2015). A 1-D radial vaneless diffuser modelaccounting for the effects of aerodynamic blockage. 1-12. Paper presented at International TurbochargingSeminar 2015, Tianjin, China.

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Download date:13. Jul. 2018

1

Abstract The radial vaneless diffuser, though comparatively simple in

terms of geometry, poses a significant challenge in obtaining

an accurate 1-D based performance prediction due to the

swirling, unsteady and distorted nature of the flow field.

Turbocharger compressors specifically, with the ever

increasing focus on achieving a wide operating range, have

been recognised to operate with significant regions of

spanwise separated flow, particularly at off-design

conditions.

Using a combination of single passage Computational Fluid

Dynamics (CFD) simulations and extensive gas stand test

data for three geometries, the current study aims to evaluate

the onset and impact of spanwise aerodynamic blockage in

radial vaneless diffusers, and how the extent of the blocked

region throughout the diffuser varies with both geometry

and operating condition. Having analysed the governing

performance parameters and flow phenomena, a novel 1-D

modelling method is presented and compared to an existing

baseline method as well as test data to quantify the

improvement in prediction accuracy achieved.

Nomenclature A Flow area (m

2)

AR Area ratio of diffuser (-)

b Passage height (m)

B Blockage (-)

Cf Skin friction coefficient (-)

CP Static pressure recovery coefficient (-)

D Diameter (m)

I Ideal / Isentropic

k Skin friction constant (-)

ṁ Mass flow rate (kg/s)

p Static pressure (Pa)

p0 Total pressure (Pa)

PR Total-total pressure ratio (-)

r Radius (m)

R Gas constant (J/kgK)

Re Reynolds number (-)

RR Radius Ratio of diffuser (-)

T Temperature (K)

T0 Total temperature (K)

U Blade speed (m/s)

V Absolute velocity (m/s)

V Volumetric flow rate (m3/s)

YD Diffuser loss coefficient (-)

γ Ratio of specific heats (-)

α2 Mean impeller tip flow angle relative to radial (°)

ϕ Local flow coefficient (-)

η Isentropic total-total efficiency (-)

ρ Density (kg/m3)

ϟ Diffuser inlet flow parameter (-)[ϟ =��√𝛾𝑅𝑇01

𝑈22𝐷2

2 ]

β Flow angle relative to meridional (deg)

1-D One-dimensional

SFM Swirl flow meter

VLD Vaneless diffuser

Subscripts:

i Calculation step

I Ideal

r Radial direction

TT Total –to-total

u Tangential direction

1 Stage inlet

2 Impeller exit / vaneless diffuser inlet

3 Vaneless diffuser exit

4 Volute exit measurement plane

max Maximum value for a given parameter

1. Introduction The radial vaneless diffuser, though comparatively simple in

terms of geometry, poses a significant challenge in obtaining

an accurate 1-D performance prediction due to the highly

swirling, distorted and unsteady nature of the flow field

emanating from the impeller. The distorted nature of the

flow has constituents in both the pitchwise and spanwise

directions, arising from the characteristic jet-wake impeller

exit flow field and the curvature of the shroud wall

respectively. Clearly, accounting for these flow features is

of paramount importance if a clear indication of

performance is to be obtained, particularly at off-design

conditions.

A recent focus on the characterisation of impeller

recirculation by Harley et al. [1] demonstrated the

propensity of modern automotive turbochargers to operate

with significant regions of recirculation at the inlet, leading

to, among other impacts, significant regions of aerodynamic

blockage being presented to the incoming flow. Impeller

exit recirculation too has received some attention in recent

years, with Qiu et al. [2] providing a means of

A 1-D RADIAL VANELESS DIFFUSER MODEL ACCOUNTING FOR THE

EFFECTS OF AERODYNAMIC BLOCKAGE

C. Stuart1*

, S.W. Spence1, S. Kim

1, D. Filsinger

2 & A. Starke

2

1 School of Mechanical and Aerospace Engineering, Queen’s University Belfast, UK

2 IHI Charging Systems International GmbH, Germany

*corresponding author: Tel.:+4428-9097-4569; fax: +4428-9097-4178

E-mail address:cstuart05@qub.ac.uk

2

characterising the recirculation present in the blade to blade

plane. As is frequently the case with centrifugal compressor

aerodynamics, research relating to the impeller has led the

way. The current work however aims to redress the balance

and evaluate the impact of flow separation and recirculation

and the associated spanwise aerodynamic blockage provided

to the flow in radial vaneless diffusers.

Previous work by the authors [3] relating to an evaluation

of existing 1-D vaneless diffuser modelling methods for

turbocharger centrifugal compressor applications

highlighted the importance of accounting for the presence of

spanwise aerodynamic blockage at the entrance to the

diffuser when specifying the flow field. An approach

involving the use of single passage CFD simulations for

each of the compressor stages in question was utilised to

permit a correlation capturing the influence of geometry and

operating condition to be generated. A new diffuser inlet

flow parameter, as depicted in Eq. (1), also had to be

specified in order to allow the variation of diffuser inlet

blockage with both geometry and operating condition to be

captured within a representative non-dimensional parameter

that could be applied to any data set. The resulting flow

parameter, which captures the dominant diffuser inlet flow

variables identified by Cumpsty [4] equates to the quotient

of local flow coefficient and impeller tip Mach number,

henceforth represented by ϟ.

ϟ =��√𝛾𝑅𝑇01

𝑈22𝐷2

2 (1)

The resulting empirical relationship between diffuser inlet

blockage B2 and the diffuser inlet flow parameter ϟ

amounted to a quartic fit line, as shown by Eq. (2).

𝐵2 = 91900ϟ4 − 62100ϟ3 + 15500ϟ2 − 1730ϟ + 90.6

(2)

The characterisation work pertaining to diffuser inlet

aerodynamic blockage demonstrated the presence of

significant levels of blockage (up to 60%), particularly

towards the surge side of the map. Specification of this

parameter permitted the complete flow field at inlet to the

diffuser to be calculated from gas stand test data gathered at

QUB, allowing an evaluation of different existing meanline

diffuser modelling methods to be undertaken.

The resulting modelling evaluation highlighted that the use

of an equivalent skin friction coefficient (Cf) as a bulk loss

term in 1-D diffuser modelling can, when tuned correctly,

deliver a performance prediction within an acceptable

window of accuracy. However, such a method does not

permit the designer to interrogate the results from the model

and easily identify the predominant sources of loss, making

it of limited use as a design tool.

The current work focuses on quantifying the degree of

spanwise separation of the flow, and how this is related to

geometry and operating condition. While it would be ideal

to directly evaluate the extent of the resulting aerodynamic

blockage presented to the flow from test data, the scale of

the stages being investigated does not permit the necessary

instrumentation to be reasonably incorporated. The

alternative approach applied for the current study was the

use of single passage CFD simulations for each geometry;

the validation of each against gas stand test data is presented

in a subsequent section.

Comparisons based on the resulting performance prediction

were drawn between the proposed modelling method, the

baseline Herbert [5] vaneless diffuser model and gas stand

test data. Three modern automotive turbocharger centrifugal

compressors, denoted C-4, C-5 and C-6 respectively, were

used in this investigation. All three compressors utilized

vaneless diffusers and backswept impeller blading,

indications to the dimensions of which are depicted in Table

1. It is worth emphasising that all three geometries are

typical of automotive stages, being devoid of recirculating

casing treatments and pre-swirl vanes. The baseline

geometry (C-4) was designed for an automotive gasoline

engine of 2.0L swept volume.

Table 1: Tested compressor stage geometries

ΔD2 (%) Δb2 (%) Δ(D3/D2) (%)

C-4 - - -

C-5 +30.8 +24.3 -8.33

C-6 +48.7 +25.1 -6.81

2. Modelling The modelling work undertaken for the current study comes

under two headings. Firstly, single passage CFD simulations

to allow the blocked region to be evaluated and the impact

on the flow field to be determined. Secondly, 1-D radial

diffuser modelling, illustrating how the findings from the

CFD study were implemented into a new modelling method.

2.1 CFD Methodology

The chosen package for conducting all CFD simulations

within the current work was ANSYS CFX14.0. In order to

balance the requirements for calculation time and modelling

accuracy, the approach taken was to employ single passage

simulations rather than a full stage calculation. While this

neglects the presence of the scroll volute as is effectively

universally found on turbocharger compressors, it will be

shown that when coupled with the correct post processing

techniques, the results from the single passage simulations

will demonstrate a satisfactory degree of accuracy.

The modelling configuration applied for each of the

compressors followed that as described by Harley et al. [1],

as illustrated in Figure 1. The single passage model was

defined to contain three separate domains, two stationary

domains for the inlet and diffuser respectively, and one

rotating domain for the impeller. As shown in Figure 1, the

total cell count was approximately 1.6 million, a value

arrived at through having conducted a grid independence

study. The Shear Stress Transport (SST) turbulence model

was employed, necessitating y+ to be maintained below five

in all three domains, with a level of less than two being

3

achieved throughout the majority of the model. The

measurement planes (MP) depicted in Figure 1 correspond

with those available from test data and with 1-D interstage

measurement points.

Figure 1: Single passage CFD setup [3]

In terms of solver convergence criteria, convergence was

deemed to have been achieved when the RMS residuals fell

below 1x10-4

[6], the imbalances of mass, energy and

momentum across the model fell below 0.01%, and the

total-to-total isentropic efficiency fluctuations were less than

0.05%. When simulating low mass flow conditions, the

surge point was defined as when the solver failed to meet

the convergence criteria.

In order to better replicate the real compressor stage,

additional loss terms had to be applied to account for aspects

not represented within the single passage models. During the

post processing of the CFD data two additional 1-D losses

were applied, namely the volute loss model of Weber and

Koronowski [7], and the disk friction loss of Whitfield [8].

The resulting CFD predictions are compared with test data

in the next section.

2.2 CFD Validation

In order to establish confidence in the accuracy of the CFD

predictions, the CFD predicted compressor maps of both

total pressure ratio and total-to-total isentropic efficiency

were compared to those gathered during gas stand testing in

QUB. The layout of the test facility is depicted

schematically in Figure 2, with details of the testing

procedure employed described in [3].

Figure 2: Schematic of QUB turbocharger test

facility [3]

It is worth emphasizing that the CFD results presented have

been post processed to include 1-D loss correlations for the

volute and disk friction, as detailed in the previous section.

The resulting comparisons between the single passage CFD

results and test data are presented in Figure 3 to Figure 5 for

C-4 to C-6 respectively.

Figure 3: Comparison of CFD results with test data for C-4

Figure 4: Comparison of CFD results with test data for C-5

4

Figure 5: Comparison of CFD results with test data for C-6

Upon inspecting the results depicted in Figure 3 to Figure 5,

it is apparent that the CFD has predicted performance within

an acceptable window of accuracy. The pressure ratio and

efficiency predictions follow the trend of the test data well,

with the maximum variation between the data sets being

witnessed at low mass flow rates for each of the

compressors. It was demonstrated in previous work ([3])

that the test data is essentially devoid of the effects of heat

transfer, however what little impact it did have on efficiency

would be most prevalent at low speeds and low mass flows.

It is notable that surge is frequently predicted at an

unrealistically low mass flow rate when compared with the

test data, with maximum deviations of 48.4%, 17.2% and

50.7% being witnessed for C-4 to C-6 respectively. It is of

course recognised however that surge is a system

phenomenon that is also fundamentally unsteady; the CFD

model used did not represent the full system and assumed

steady flow since surge prediction was not a primary target

of the modelling work. In addition, for C-4 there were

convergence issues which could not be resolved within the

upper two speed lines, yielding a substantially truncated

map prediction at these operating conditions. These

discrepancies are not entirely unexpected with a single

passage simulation however. While a 1-D volute loss model

has been applied, it is not sufficient to capture the non-

axisymmetric, three-dimensional and unsteady nature of the

volute flow field, and its impact on the performance and

stability of components upstream at off-design conditions

[9].

In fact, the prediction of surge to be at differing flow rates to

the test data does not pose a problem for the current work.

The blockage values used in a subsequent section, which

represent the full extent for which CFD was used in the

current work, were only extracted at mass flow rates that fall

within the bounds of the operating range as defined by the

test data. Therefore, with the magnitude of the CFD

predictions having been shown to be reasonable within the

confines of the test data, the methodology applied is fully

sufficient for the current study.

2.3 Characterisation of Diffuser Blockage

Having gained confidence in the accuracy of the CFD

simulations, it was possible to move forward with the 1-D

modelling work. In order to evaluate the radial extent of this

aerodynamic blockage, the same approach was applied as

was used in the diffuser inlet blockage study referred to in a

previous section, but instead of being evaluated exclusively

at diffuser inlet, it was applied at approximately 25 discrete

radial locations from the inlet to exit of the diffuser

(depending on geometry). Again, the Turbo Chart feature of

ANSYS CFX was utilized to extract circumferentially

averaged streamwise velocity and density values at 250

discrete points equally spaced from hub to shroud at each of

the chosen streamwise locations for each compressor.

Using the continuity equation, it was then possible to

calculate the mass flow passing through each pitchwise

“slice” of the diffuser passage (denoted “dz” in Figure 6).

The extent of the active flow region was then calculated by

summing from hub to shroud until the stage mass flow rate

was reached. The proportion of the remaining diffuser slices

in relation to the total number of 250 defined the extent of

the aerodynamic blockage. A schematic representing this

procedure is depicted in Figure 6.

Figure 6: Determination of diffuser aerodynamic blockage

The result of this analysis is a diffuser passage that is

defined by two separate regions; an active flow region

through which the entirety of the stage mass flow is

assumed to pass, and a blockage region which makes no

contribution to outlet flow conditions. The aerodynamic

blockage region influences the flow field in three ways:

increased radial velocity component VR2 (by

continuity)

reduction in absolute flow angle α2

modification of effective diffuser inlet to outlet

area ratio

The first two parameters directly impact upon the flow field,

while the final one has an indirect impact on actual diffuser

5

performance through modifying the ideal achievable CP for

the diffuser [10], as illustrated in Eq. (3). As will become

apparent in the coming sections, this final element has a

significant impact on diffuser performance.

𝐶𝑃𝐼 = 𝑐𝑜𝑠2(𝛼2) (1 −1

𝐴𝑅2)

+ 𝑠𝑖𝑛2(𝛼2) (−1

𝑅𝑅2)

(3)

In order to capture the variation of blockage across the

different geometries and operating conditions being tested, it

was deemed necessary to formulate a simplified method

reliant on data from only a number of operating points,

rather than the full compressor map. The variation of the

blockage throughout the diffuser was extracted at surge,

choke and peak efficiency for all three geometries at a

number of operating conditions. A sample result from this

analysis is presented in Figure 7 for C-6 at 75% speed,

detailing the CFD results as well as the correlations

generated to replicate the blockage variation on a 1-D basis.

Figure 7: Diffuser blockage variation for C-6 at 75% speed

The trend illustrated in Figure 7 is one that was echoed

across all three geometries at a range of operating

conditions; the high levels of inlet blockage at surge

depicted a tendency to decrease through the diffuser, while

the comparatively low levels of inlet blockage at choke

tended to increase with radius in the diffuser. At the surge

side of the map, low velocity fluid is subject to a strong

adverse pressure gradient, aggravating the tendency for

boundary layer separation and recirculation. The

relationship between blockage and diffuser radius ratio

witnessed in Figure 7 can be attributed to the decay of this

recirculation zone present close to impeller exit, as depicted

in Figure 6. Towards choke, the fluid velocity is greater and

the adverse pressure gradient much less severe. As a result,

the recirculation region which is dominant at surge is no

longer present, meaning the blockage presented to the flow

is predominantly attributable to boundary layer growth from

inlet to outlet of the diffuser. Therefore, having analysed the

blockage levels throughout the diffuser for the three

geometries under consideration at a range of operating

conditions, it became apparent that the overall trend could

be well represented (albeit on a simplified basis) at both

surge and choke by exponential functions, as illustrated in

Figure 7.

The proposed correlations, as illustrated in Eq. (4) and (5),

represent either an exponential decay or growth of the

diffuser aerodynamic blockage for surge and choke

respectively. The exponent “x” represents radial location in

the diffuser, which equates to the quotient of the current

calculation step, i, and the total number of calculation steps.

Throughout the current analysis performance was evaluated

using 100 radial steps through the diffuser, a value that was

deemed to balance the competing aims of calculation time

and prediction accuracy.

𝐵𝑖,𝑠𝑢𝑟𝑔𝑒 = 𝐵2𝑒−2𝑥 (4)

𝐵𝑖,𝑐ℎ𝑜𝑘𝑒 = 𝐵2𝑒0.5𝑥 (5)

At this stage, representative correlations have been

generated for surge and choke, however no consideration

has been given for mid map conditions. In order to

accurately represent the blockage present away from the

extremities of map width, a linear interpolation procedure

was employed on the basis of the diffuser inlet flow angle

α2, as depicted in Eq. (6). This approach provided a smooth

transition between the two correlations and, as will become

apparent in the coming sections, a good estimation of

diffuser performance across the operating range without the

need for extensive knowledge about the flow field at every

radial location in the diffuser at each operating point.

𝐵𝑖

= 𝐵𝑖,𝑠𝑢𝑟𝑔𝑒 + (𝐵𝑖,𝑐ℎ𝑜𝑘𝑒

− 𝐵𝑖,𝑠𝑢𝑟𝑔𝑒) (𝛼2 − 𝛼2,𝑠𝑢𝑟𝑔𝑒

𝛼2,𝑐ℎ𝑜𝑘𝑒 − 𝛼2,𝑠𝑢𝑟𝑔𝑒

)

(6)

3. 1-D Diffuser Modelling Based on the findings from the previous study [3], it was

deemed that the most appropriate model to build upon for

the current work was a model based upon an equation first

presented by Rodgers [11], but subsequently converted into

a vaneless diffuser modelling method by Stuart et al. [3].

The Rodgers equation permitted evaluation of diffuser CP

directly, knowing only the overall diffuser geometry and

inlet flow angle. This was further developed to calculate

diffuser pressure recovery for each radial step, as well as the

associated key flow parameters using a compressible

analysis.

The reasoning behind this was that, ultimately, the aim was

to develop a vaneless diffuser model for which each of the

individual sources of loss were accounted for, and with the

simplistic method outlined above only currently accounting

for the impact of wall friction, it was deemed a good basis to

build upon. In order to evaluate the improvement in

prediction over existing methods, the Herbert model [5],

which is an extension and correction of Stanitz’ [12] ground

breaking work, was chosen as it represented the most

6

complex of the existing approaches, incorporating terms for

boundary layer growth and total pressure loss as a function

of Mach number and boundary layer shape factor.

Fundamentally, the proposed model evaluates the coefficient

of static pressure rise, CP, for each calculation step in the

diffuser, based upon geometry and a chosen value of skin

friction coefficient Cf using the aforementioned Rodgers

equation [11], as depicted in Eq. (7). Unlike a number of

existing approaches which effectively employ the skin

friction coefficient as a bulk loss term, it is intended that that

the variable in the proposed model accounts only for wall

friction loss.

𝐶𝑃 = [1 − (𝐷(𝑖)

𝐷(𝑖+1)

)

2

]

−𝐶𝑓

cos (𝛼𝑖)

𝐷(𝑖)

𝑏(𝑖)

(1 −𝐷(𝑖)

𝐷(𝑖+1)

) (7)

From this, the ideal isentropic coefficient of static pressure

rise, CPI, is evaluated using Eq. (3), and compared with the

actual value calculated using Eq. (7) to allow the vaneless

diffuser loss coefficient YD to be calculated, as shown in Eq.

(8).

𝑌𝐷 = 𝐶𝑃𝐼 − 𝐶𝑃 (8)

Knowing the value of YD, it is possible to calculate the

change in total pressure across the current calculation step,

and hence evaluate the total pressure applicable to the

following step, as illustrated in Eq. (9) and Eq. (10)

respectively.

𝑑𝑝𝑜 = 𝑌𝐷(𝑝𝑜(𝑖) − 𝑝(𝑖)) (9)

𝑝𝑜(𝑖+1) = 𝑝𝑜(𝑖) − 𝑑𝑝𝑜 (10)

Before moving on to calculate the remaining flow variables,

it is necessary to incorporate the impact of the previously

described aerodynamic blockage on the flow field. In order

to incorporate the blockage correlations into a 1-D

modelling method, the direct impact on the flow field had to

be evaluated. The first parameters requiring evaluation for

each calculation step were the diffuser inlet flow parameter

ϟ, and the associated quartic relationship for diffuser inlet

blockage B2, as depicted in Eq. (1) and (2) respectively.

Knowing these parameters, and the exponent x, it is possible

to evaluate the surge and choke blockage relationships given

by Eq. (4) and (5), and ultimately calculate the blockage at

any step using the linear interpolation based on absolute

diffuser inlet flow angle α2 detailed in Eq. (6).

Knowing the blockage presented to the flow, it was possible

to evaluate the direct impact on the radial velocity using the

continuity equation, as illustrated in Eq. (11).

𝑉𝑟𝑖 =��

𝜌𝑖(1 − 𝐵𝑖)(2𝜋𝑟𝑖𝑏𝑖) (11)

As the analysis is compressible in nature, a further internal

iterative process was incorporated into the analysis to

determine the density at each step. This involved

undertaking the above procedure, then continuing the

calculation to determine the total and static pressure and

temperature values associated with the newly calculated

flow conditions. From the resulting static pressure and

temperature values, a new density was calculated using the

equation of state, and compared with the initial value. If

satisfactory convergence between the two density values

was not achieved, the entire calculation was completed

again, beginning with the determination of the blockage

level, until such a point where the solution was deemed to

have converged.

Having specified the basis of the proposed modelling

method, it was possible to evaluate its impact on the

prediction of diffuser performance. The parameter upon

which the modelling methods will be evaluated is the

coefficient of static pressure rise (CP). CP, as depicted in

Eq. (12), captures the fundamental purpose of a centrifugal

compressor diffuser, by providing a metric to determine the

proportion of total pressure at inlet of the diffuser that was

successfully converted into static pressure at the exit of the

diffuser.

𝐶𝑃 =𝑝3 − 𝑝2

𝑝02 − 𝑝2

(12)

As a first step, a direct comparison of diffuser CP prediction

was undertaken for each of the geometries. This utilised the

same methodology as was applied by Stuart et al. [3], where

diffuser inlet conditions were specified entirely from testing

data to ensure the integrity of the comparison and to remove

any discrepancies attributed to limitations in the meanline

impeller modelling. The resulting comparison between test

data, the Herbert model and the proposed model for each of

the three geometries is presented in Figure 8.

In order to improve the clarity of the resulting comparison,

only three speedlines (representing low, medium and high

tip speeds) for each geometry are presented. Furthermore,

the values of CP in each case have been non-

dimensionalised using the maximum value of CP achieved

across all three geometries, be that from test results or 1-D

predictions. It must be noted that this approach is in

contravention of what was conducted for the other figures

within the manuscript, where the maximum value for a

given parameter was taken for each compressor

individually. However, it helps to illustrate a key

observation within the discussion that follows.

7

Figure 8: 1-D diffuser modelling comparison

Discussion What is immediately apparent from analysing Figure 8 is

that the baseline Herbert model is lacking the ability to

predict the trend of how CP varies with α2 across the

compressor map. Despite the relative complexity of the

modelling method, as explained in the preceding sections,

the predominant feature of the approach is the impact of the

skin friction coefficient Cf. This dictates that CP variation

across the map will be mainly reliant on the relationship

between the chosen friction coefficient, and the flow path

length within the diffuser. As a result, for a given value of

Cf, CP will be maximum for the lowest value of α2, which

corresponds to the shortest flow path, and consistently

decrease as the flow angle becomes more tangential. Due to

the additional parameters in the model relating the loss in

total pressure through the diffuser to the boundary layer

shape factor, which in turn is dependent upon Mach number,

there is some spreading of the speedlines at a constant value

of α2. However, even with this addition the influence of the

friction loss is still predominant.

A further point worth noting is that it is evident that,

generally speaking, the 1-D modelling methods deliver

better correlation with the test data for C-5 and C-6 than for

C-4, which has the smallest impeller but the largest diffuser

radius ratio of the three test cases. It is readily apparent from

Figure 8 therefore that the increase in the ideal CP, as

depicted in Eq. (3), brought about by the larger radius ratio

(RR) of C-4 is not sufficiently counteracted by the increased

frictional losses associated with the longer flow path for a

given increase in flow angle within the diffuser.

With respect to the proposed modelling method, it is clear

that it offers an improved prediction, particularly at the

extremes of the compressor operating map. At higher values

of α2 the approach of directly applying the loss in total

pressure associated with the skin friction coefficient in each

geometry step delivers a more accurate representation of

real performance when compared with the Herbert model,

illustrated by the substantial performance decrement

towards surge. For example, considering the maximum α2

values reached by C-5 for each of the three speedlines under

consideration, the maximum deviation from the test data

witnessed with the proposed model was 1.0% compared

with 25.7% for the Herbert model. Similarly, for C-6 the

largest discrepancy with the test data witnessed was 13.6%

compared to a value of 43.6% with the Herbert model. As

described above, the correlation achieved for C-4 was not as

impressive as for the other two geometries, however the

proposed method still delivers a more representative

prediction than that achieved by the Herbert model.

At the opposite side of the performance map, the

performance decrement towards choke that is evident in the

test data, which is absent in the Herbert prediction, is

captured well by the proposed model, especially for C-5 and

C-6. Comparing again the maximum deviation between the

test data and each of the modelling methods, this time at

minimum α2, for C-5 this amounted to 7.0% for the

proposed model, compared to 22.2% for the Herbert model.

Similarly, for C-6 the maximum deviation of 12.9% for the

proposed model compares favourably with the 48.6%

arising from the Herbert model. In the same vein as

previously mentioned, the prediction from the proposed

method for C-4 did not follow as well as for the other two

geometries, but still offered an improvement over the

baseline Herbert method. While the reasoning behind this

has been covered in the preceding paragraphs, further work

would be required to gain an understanding of the best way

to modify the model to account for the difference in

performance for C-4.

In terms of an explanation for the drop off in performance

towards choke (low values of α2) within the proposed

model, the blockage correlation for choke as depicted in Eq.

(5) dictates that from inlet to outlet, the level of blockage

presented to the flow must increase. As a result, the

8

effective geometry of the diffuser has been modified,

meaning the area increase associated with each calculation

step is much less than would be expected taking only the

physical geometry into account. A simplified schematic

illustrating this point are depicted in Figure 9.

Figure 9: Schematic of diffuser geometry with blockage

As is illustrated in Figure 9, while the levels of blockage at

the inlet of the diffuser are much smaller at choke than at

surge, it is the growth of the blocked region towards choke

that causes the significant decrement in performance. As the

effective passage height is reducing for each calculation

step, the area increase associated with the increase in radius

is diminished over the case with no blockage, or the surge

case. Going back to Eq. (3), the reduction in the area ratio of

the diffuser brings about a reduction in the ideal achievable

CP, resulting in the ability to replicate the performance

decrement witnessed in the test data. It is clear therefore

from the current model that wall friction is still the

predominant 1-D source of loss towards surge, but

aerodynamic blockage throughout the diffuser has been

identified to be a significant contributing factor towards the

choke side of the map.

3.1 Impact on 1-D Stage Calculation

As a final verification of the benefit of the proposed

vaneless diffuser model, the decision was taken to evaluate

the impact on a 1-D stage performance prediction,

incorporating models for the impeller, vaneless diffuser and

volute. The impeller losses applied to the model originated

from the work of Galvas [13], with the inlet recirculation

model of Harley et al. [1] and the choking loss of Aungier

[14] was also applied. The slip factor correlation applied

was that of Qiu et al. [15], with the volute loss again being

accounted for using the work of Weber and Koronowski [7].

As with the previous section detailing a diffuser only

performance evaluation, the vaneless diffuser model used

for comparison against the proposed model was that of

Herbert [5].

The resulting comparisons for C-4 to C-6 are illustrated in

Figure 10 to Figure 12 respectively, where all data have

been normalised using the maximum respective value for

that parameter. As with the previous section, to preserve the

clarity of the comparison only three speed lines per

geometry have been presented, representing low, medium

and high tip speed operation.

Figure 10: Comparison of 1-D modelling with test data for

C-4

Figure 11: Comparison of 1-D modelling with test data for

C-5

CHOKE SURGE

9

Figure 12: Comparison of 1-D modelling with test data for

C-6

Discussion

What is instantly striking about the comparisons illustrated

in Figure 10 to Figure 12 is that the magnitude of the

improvement in the prediction achieved on a diffuser only

basis has not been reflected as significantly on a stage

performance basis. Generally speaking, the change in

performance associated with the proposed model has a

greater impact on the choke side of the map, however while

this is advantageous in bringing the prediction closer to the

test data at higher speeds, it generally has a detrimental

impact on the lower speed lines. This observation is

particularly pertinent for C-6, as depicted in Figure 12.

A common observation for each of the geometries is that the

accuracy of both the pressure ratio and efficiency

predictions diverge as compressor speed increases. The

trend depicted in the test data concerning the drop in peak

compressor efficiency at high speeds is one that is not

captured by the existing 1-D modelling technique, resulting

in diminishing efficiency prediction with increasing speed.

As noted by Harley et al. [1], this trend is common in

automotive turbocharger compressors, where designs are

specifically targeting improved efficiency at low speeds and

mass flow rates to better align with the engine transients

encountered during urban driving.

What is also evident is while the efficiency prediction

diverges at higher speeds, the prediction of pressure ratio

does so at an increased rate. Taking C-5 as an example, on

the 100% speedline at m/mmax of 0.75, the error in the

efficiency prediction equates to 5.6% between the test data

and proposed model, while the pressure ratio prediction

illustrated an associated error of 21.3%. Upon investigating

this problem more deeply, it became apparent that the

discrepancy was related to the prediction of the tangential

velocity at impeller exit (Vu2) from the single zone model.

Comparing the prediction of Vu2 from the single zone model

to that from the previously described CFD simulations, an

over prediction of 19.2% by the single zone model was

evident for the same operating point for C-5. It would

appear that the issue here relates to the determination of the

impeller slip factor, as if it was over predicted, it would

manifest itself as an increase in predicted pressure ratio (in

accordance with the Euler equation). However, as slip factor

is not a loss, it would not have the same impact on

efficiency, which is exactly the symptom depicted in the

current data set.

A further issue with the 1-D prediction is the prediction of

the choking mass flow rate, the accuracy of which again

diverges with increasing compressor speed. The current

method applied is that of Dixon & Hall [16], which

compares the available inducer throat geometric area to a

calculated choking area based on total inlet conditions.

Unfortunately, this simplified approach cannot account for

the highly non-uniform velocity profile at the throat section

and the associated boundary layer blockage [17], which like

in the current work, results in the full geometric area not

being available to pass the stage mass flow. Consequently,

choking in the real stage occurs significantly earlier than

what is predicted by the simplistic 1-D analysis currently

applied. Again, further work is required to improve the

fidelity of the choking model to account for these real flow

effects.

4. Conclusions Improvements in the 1-D prediction of vaneless diffuser

performance have been achieved through the analysis of

three automotive turbocharger centrifugal compressor

stages. A combination of extensive gas stand test data,

which was shown to have been gathered under

approximately adiabatic conditions, as well as single

passage CFD simulations for each geometry were employed

to permit full characterisation of the diffuser flow field.

Modelling of the extent of the aerodynamic blockage

presented to the flow throughout the diffuser with changing

geometry and operating conditions was completed on a

simplified basis, and was illustrated to have a significant

impact on diffuser performance. Correlations describing the

variation in the aerodynamic blockage throughout the

diffuser resulting from the stratified flow field emanating

from the impeller were developed, and incorporated into a

new diffuser modelling method described herein.

Utilising the methodology developed by the authors for a

previous study, direct comparisons were drawn between the

diffuser performance prediction delivered by the proposed

model, the baseline 1-D Herbert model and test data. By

providing the 1-D models and test data with effectively

common input parameters, a robust diffuser-only

performance analysis was conducted for each of the

geometries, validating significant improvements in the off-

10

design performance prediction delivered by the proposed

model. This improved performance prediction for vaneless

diffusers is certainly beneficial at the preliminary design

stage, allowing the designer to consider appropriate

geometry before committing to 3D CFD modelling.

Upon incorporating the proposed diffuser model into a 1-D

stage calculation, the limitations in the modelling of the

other stage elements (particularly the impeller) masked the

benefits witnessed on a diffuser-only basis. Comparing the

1-D stage prediction with test data for each geometry did

however provide some guidance for future work, with the

divergence in pressure ratio and efficiency prediction

towards higher speeds and the overall choking mass flow

prediction providing focal points.

Acknowledgements The authors would like to sincerely thank IHI Charging

Systems International GmbH for provision of the necessary

compressor hardware for testing, as well as for their

continuing technical support. The authors would also like to

extend their thanks to ANSYS Inc. for the use of their CFD

software and their technical support during this programme

of research.

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